Find ∏2∞ (n3 - 1)/(n3 + 1).
Solution
Answer: 2/3.
If we factor: n3 - 1 = (n - 1)(n2 + n + 1), n3 + 1 = (n + 1)(n2 - n + 1), then most of the terms cancel. Take the product up to n = N. Then the numerator is 1·2·3 ... (N - 1)·7·13·21·31 ... (N2 + N + 1) and the denominator is 3·4·5 ... (N + 1)·3·7·13·21·31 ... (N2 - N + 1). Hence the product up to n = N is 2/(N(N + 1)) (N2 + N + 1)/3 = 2/3 (N2 + N + 1)/(N(N + 1)), which tends to 2/3 as N tends to infinity.
© John Scholes
jscholes@kalva.demon.co.uk
30 Nov 1999